MA thesis available - University of Hawaii

6 M. CHI
(1) We have
aτ + b
f = (cτ + d)
cτ + d
k f(τ),
a b
for all τ ∈ H and all ∈ Γ.
c d
(2) If γ ∈ SL2(Z), then (f|kγ)(τ) has a Fourier expansion of the form
(f|kγ)(τ) =
aγ(n)q n N,
Remark 1.
where qN := e 2πiτ
N and aγ(nγ) = 0.
n≥nγ
(1) Condition (2) of Definition 3 means that f(τ) is meromorphic at the cusps of Γ. If
nγ ≥ 0 for each γ ∈ SL2(Z), then we say that f(τ) is holomorphic at the cusps of Γ.
(2) Condition (1) may be rewritten as (f|kγ) = f for all γ ∈ Γ.
(3) By [13, III.3 Modular forms for congruence subgroups], the function f|kγ is periodic
with period N for γ ∈ SL2(Z)/Γ.
(4) One uses only even integers k in the definition above since it is easy to check that
there are no non-zero modular forms of odd weight.
Definition 4. Suppose that f(τ) is an even integer weight meromorphic modular form on
a congruence subgroup Γ0(N).
We say that f(τ) is a holomorphic modular form if f(τ) is holomorphic on H and is
holomorphic at the cusps of Γ0(N).
We say that f(τ) is a holomorphic cusp form if f(τ) is holomorphic on H and vanishes
at the cusps of Γ0(N).
We say that f(τ) is a weakly holomorphic modular form if it is holomorphic on H
with possible poles in cusps of Γ0(N).
When Γ = Γ0(N) is a subgroup of SL2(Z), we put
Mk(Γ) = { f: f is a holomorphic modular form of weight k on Γ},
Sk(Γ) = { f: f is a cusp form of weight k on Γ},
M ! k (Γ) = { f: f is a weakly holomorphic modular form of weight k on Γ}.
Let q = e2πiτ with τ ∈ H . One of the first classical examples of a cusp form is
∆ = q
(1 − q n ) 24 ∈ S12(Γ0(1)).
n≥1
Mk(Γ) and Sk(Γ) are finite dimensional vector spaces. M ! k (Γ) is an infinite dimensional
vector space. Sk(Γ) has a basis {gi} such that gi = ∞ n=1 ci(n)qn with ci(n) ∈ Z. The
latter fact is rather difficult to prove, and we take it for granted. This fact has several
implications. Let f =
n≫−∞ a(n)qn ∈ M ! k (Γ) be a modular form such that all coefficients
a(n) are algebraic numbers. Then there is an algebraic number field K (i.e. a finite extension
of Q) such that a(n) ∈ K for all n. Moreover, there exists T ∈ K such that T a(n) ∈ OK, the
ring of integers of K, for all n. The latter statement is often called “bounded denominators